Chapter 7 Week 8 EE 341 Clark Lecture Notes - 20 may

Chapter 7 Week 8 EE 341 Clark Lecture Notes - 20 may - EE...

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EE 341 Discrete-Time Linear Systems – Week 8 Spring 2011 1 Text reading assignment for this week: Sampling: Chapter 7, Sec 0-3 Review LTI eigenfunctions and eigenvalues: Chapter 3, Sec 2. Magnitude and phase response: Chapter 6, Sec 2. LINEAR CONVOLUTION IN TIME As usual,  [] [] [] F jj j yn xn hn Y e X e H e   Ex. Given [] [] 1 n hn aun a  . Find its inverse system [ ] i .
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EE 341 Discrete-Time Linear Systems – Week 8 Spring 2011 2 Class Problem: Find the inverse DTFT of   2 sin 3 j Xe .
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EE 341 Discrete-Time Linear Systems – Week 8 Spring 2011 3 MULTIPLICATION OF SIGNALS 12 1 2 1 [] [] ( ) 2 jj x nx n X e  where this denotes CIRCULAR CONVOLUTION in frequency, defined as: () 1 2 2 ()( ) j j X eX e X e X e d  [] T a k e i t s DTFT yn n j n nn Ye yne xnx ne      11 2 1 2 ja jan x nX e e d a 22 2 1 2 jb jbn x e e d b j ja jan jb jbn j n n X e e da X e e dbe      2 1 ()() 2 ja jb j a b n j n n X e X e e e dadb     Now, ja bn j n n ee is just the DTFT of jabn e that is, 2( ) F ea b   So, 1 2 2 1 ( ) 2 ) a j b ja j a j j Y e X e X e a b dadb X eXe d a Xe  
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EE 341 Discrete-Time Linear Systems – Week 8 Spring 2011 4 Ex. 1: The frequency-shift property is a special case where 1 [] x nx n and 0 2 jn x ne . Ex. 2: Truncated sinc function,   sin [] [] Wn hn rn n , where [] 0f o r n L . -2 0 2 0 5 10 15 20 -3 -2 -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 -2 0 2 0 5 10 15 20 -3 -2 -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1
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EE 341 Discrete-Time Linear Systems – Week 8 Spring 2011 5 Review of Sampling Continuous Time Signals and Connection to the Fourier Transform of Periodic Signals (Slightly different derivation, yet with the same notation as text Section 7.1) In practical systems sampling is used, for example, in A/D conversion. For EE 341 we ignore the effects of amplitude level quantization and instead focus on sampling in time.
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Chapter 7 Week 8 EE 341 Clark Lecture Notes - 20 may - EE...

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